4.2 Linear Approximations and Differentials

Linear approximations and differentials are powerful tools in calculus for estimating function values. They use tangent lines to approximate curves near specific points, providing a simpler way to understand complex functions.

These techniques are crucial for solving real-world problems where exact calculations are impractical. By understanding linear approximations and differentials, you'll gain insights into function behavior and improve your problem-solving skills in calculus.

Linear Approximations and Differentials

Linear approximation at a point

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• Estimates the value of a function near a specific point using a linear function (the tangent line to the curve at that point)
• Tangent line provides a good approximation of the original function near the point of tangency (local approximation)
• Based on the idea that a differentiable function can be closely approximated by a linear function near a given point due to the proximity of the tangent line and the curve in that vicinity

Construction of function linearization

• Linearization of a function $f(x)$ at a point $a$ given by the formula: $L(x) = f(a) + f'(a)(x - a)$
  • $f(a)$ value of the function at the point of linearization
  • $f'(a)$ derivative of the function at the point of linearization
  • $(x - a)$ difference between the input value and the point of linearization
• $L(x)$ represents the equation of the tangent line to the curve of $f(x)$ at the point $(a, f(a))$
• Linearization used to estimate the value of $f(x)$ for $x$ near $a$
  • Example: $f(x) = \sin(x)$, estimate $\sin(0.1)$ using linearization at $a = 0$
    1. $L(x) = \sin(0) + \cos(0)(x - 0) = x$
    2. $L(0.1) = 0.1$, good approximation of $\sin(0.1) \approx 0.0998$
• The slope of the linearization represents the instantaneous rate of change of the function at the point of linearization

Graphical representation of differentials

• Differentials estimate the change in a function's value based on a small change in its input
• Function $y = f(x)$, differential $dy$ represents a small change in $y$ corresponding to a small change $dx$ in $x$
  • $dy$ approximated by the product of the derivative $f'(x)$ and $dx$: $dy \approx f'(x) \, dx$
• Graphically, $dy$ represented as the vertical change between the tangent line and the original function curve at a given point
  • Tangent line (linear approximation) used to estimate the change in the function's value
  • Actual change is the vertical distance between the original curve and the point on the curve corresponding to the new input value

Error analysis in differential approximations

• Important to understand the accuracy of differential approximations
• Absolute error: difference between actual change in function's value and estimated change using the differential
  • Absolute error = |Actual change - Estimated change|
• Relative error: ratio of absolute error to actual change
  • Relative error = (Absolute error) / (Actual change)
• Percentage error: relative error expressed as a percentage
  • Percentage error = (Relative error) × 100%
• Interpreting errors helps understand the accuracy of the differential approximation
  • Smaller relative or percentage error indicates a more accurate approximation
  • Acceptable level of error depends on context and desired precision

Continuity and Differentiability

• Continuity is a prerequisite for differentiability
• A function is differentiable at a point if it is continuous at that point and has a well-defined derivative
• Differentiability implies that the function can be approximated by a linear function (tangent line) near the point of interest